Stability analysis through folds: An end-loaded elastic with a lever arm

Many physical systems can be modelled as parameter-dependent variational problems. The associated equilibria may or may not exist realistically and can only be determined after examining their stability. Hence, it is crucial to determine the stability and track their transitions. Generally, the stability characteristics of the equilibria change near folds in the parameter space. The direction of stability changes is embedded in a specific projection of the solutions, known as distinguished bifurcation diagrams. In this article, we identify such projections for variational problems characterized by fixed-free ends -- a class of problems frequently encountered in mechanics. Using these diagrams, we study an Elastica subject to an end load applied through a rigid lever arm. Several instances of snap-back instability are reported, along with their dependence on system parameters through numerical examples. These findings have potential applications in the design of soft robot arms and other actuator designs.